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algebraic technique can give the voltages and currents for any circuit. We
illustrate this latter case in Chapter 8.
Recalling that the voltage drops across resistors, inductors, and capacitors
can all be expressed as function of the current, its derivative, and its integral,
our goal is to find a technique to replace these operators by simple algebraic
operations. The key to achieving this goal is to realize that:

jφ
If: I = I cos(ω t + )φ = Re[ e jtω ( I e )] (6.80)
0 0

dI jtω jφ
Then: =− I ω sin( ω t + φ) = Re[ e ( I jω( e ) )] (6.81)
0
0
dt
and

∫ Idt = I ω 0 sin( ω t + φ) = Re    e  jtω   0  1    e jφ     (6.82)
I


j  ω


From Eqs. (4.25) to (4.27) and Eqs. (6.80) to (6.82), we can deduce that the pha-
sors representing the voltages across resistors, inductors, and capacitors can
be written as follows:

˜
˜
V = IR = ˜ IZ (6.83)
R R
V = ( ˜
˜
˜
I j L) =ω
L IZ L (6.84)
˜
˜
V = I ˜ = IZ (6.85)
jC)
C ( ω C
The terms multiplying the current phasor on the RHS of each of the above
equations are called the resistor, the inductor, and the capacitor impedances,
respectively.

6.8.1 RLC Circuit Phasor Analysis
Let us revisit this problem first discussed in Section 4.7. Using Kirchoff’s volt-
age law and Eqs. (6.83) to (6.85), we can write the following relation between
the phasor of the current and that of the source potential:

˜ I  1 
˜
˜
j L +
V = IR + ˜ ( I j L + = ˜ IR + ω  (6.86)
ω

)
jC)
ω
s ( ω  jC 
© 2001 by CRC Press LLC